Bond-dependent symmetric and antisymmetric superexchange interactions in La${}_2$CuO${}_4$

The effective spin Hamiltonian that describes the magnetic properties of La${}_2$CuO${}_4$ is derived from the nearest-neighbor superexchange interactions. It is shown that the spin system of the CuO${}_2$ plane of the compound is frustrated; the principal axes of the symmetric parts of the one-bond anisotropy tensors are not the same for all the bonds. We also show that, due to a hidden symmetry of the one-bond anisotropic superexchange interactions, the symmetry of the lattice alone leads to the identification of the largest eigenvalue of the mean-field superexchange anisotropy tensor. The derived mean-field spin Hamiltonian is identical to that used previously on a phenomenological basis to account for the magnetic properties of La${}_2$CuO${}_4$ in the orthorhombic phase. Similar arguments explain the magnetic structure in the low-temperature tetragonal phase. These structures cannot be obtained without the inclusion of the symmetric part of the anisotropy tensor, neglected in other papers. Finally, we show how the addition of direct exchange may explain the observed spin-wave gaps.